Simplify; express your answer in exponential form. Assume $q\neq 0, n\neq 0$. $\dfrac{{(q^{-3}n^{-4})^{-5}}}{{(q^{5}n^{-3})^{3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-3}n^{-4})^{-5} = (q^{-3})^{-5}(n^{-4})^{-5}}$ On the left, we have ${q^{-3}}$ to the exponent ${-5}$ . Now ${-3 \times -5 = 15}$ , so ${(q^{-3})^{-5} = q^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-3}n^{-4})^{-5}}}{{(q^{5}n^{-3})^{3}}} = \dfrac{{q^{15}n^{20}}}{{q^{15}n^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{15}n^{20}}}{{q^{15}n^{-9}}} = \dfrac{{q^{15}}}{{q^{15}}} \cdot \dfrac{{n^{20}}}{{n^{-9}}} = q^{{15} - {15}} \cdot n^{{20} - {(-9)}} = n^{29}$